The Principles of Derivatives- Secant Line, Tangent Line, and Power Rule

A derivative is a function that represents the instantaneous rate of change of another function. To first understand this concept, we first need to look back at the concept of slopes. The slope is a numerical value that represents the steepness of the function. The higher the value, the greater the steepness of the function. The slope can be calculated by taking the difference of two y coordinates of a function and dividing that by the difference of the two corresponding x functions. This would give you the average rate of change of the function over that interval of x-coordinates. When drawn on the graph of the function, the average rate of change of the function over that interval, or the slope over that interval, is known as the secant line. This line intersects two points on the curve in which the average rate of change was calculated. Now, back to derivatives, we just learned that a derivative is a function that represents the instantaneous rate of change of another function. The instantaneous rate of change of a function is graphically represented by a tangent line. This line is one that just touches one specific point in the graph and no other points, hence representing the slope of the point at that specific point, otherwise known as the instantaneous rate of change. How do you calculate the derivative that represents the instantaneous rates of change of another function? Well, there are multiple techniques based on the complexity of the equations. This ranges from specific techniques for calculating trig derivatives to chain rule for functions that have functions contained within them to quotient rule for functions that have both a numerator and a denominator, and product rule for functions which may be factored that may need to get multiplied together. But for the most basic functions, the power rule is all we need. The power rule states that the derivative of a function such as x^2 would be taken by multiplying the function by the value of the power, which in this case is 2, and subtracting the power by 1. This would make the derivative of x^2, 2x. Although it may be hard to figure it out at first, by continuing to practice the power rule, it will be second nature to you in no time. The concepts of derivatives are important to know because they come into play with such topics such as position, velocity, and acceleration in which velocity is the derivative of position and acceleration is the derivative of velocity. You would use this specific topic if you want to know the speed of the car or the acceleration of the car for example. Now, hopefully, you have a better understanding of the world of Calculus and how derivatives work.

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